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1 Department of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan
2 ERATO, Complex Systems Biology Project, JST
3 Department of Mathematical and Life Sciences, University of Hiroshima, 1-3-1, Kagamiyama, Higashi-Hiroshima, 739-8526, Japan
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Abstract |
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 Top Abstract IntroductionResultsDiscussionProceduresAppendixReferences |
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Introduction |
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From this viewpoint, Shen-Orr and colleagues recently found several types of small-scale structures that frequently appear in many regulatory networks (Shen-Orr et al. 2002), and named "network motifs" composed of several nodes and arrows. One of the most prominent network motifs is the feed-forward loop (Shen-Orr et al. 2002), which consists of three gene products and three transcriptional interactions, as shown as X, Y, and Z, and 1, 2, and 3, respectively in Fig. 1A,B. Transcription factor X regulates both Y and Z, and Y regulates Z. Thus, X modulates Z expression by two pathways, a direct pathway (X
Z), and an indirect pathway mediated by Y (XYZ). The effect of a transcriptional interaction is either positive (activation) or negative (repression). Therefore, the overall effect of the indirect pathway mediated by Y on Z depends on the regulatory effects of interactions 1 and 3. The effects of X and Y on Z are integrated at the promoter region of the gene that encodes Z. The dependence of the level of Z on the concentrations of X and Y is described by a cis-regulatory function of Z (Buchler et al. 2003; Setty et al. 2003).
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The network motifs do not often represent independent units that are functionally isolated from the rest of the network. Several network motifs interact or cross talk with each other to form a complex structure. The functioning of the network may be cooperatively generated by the cross talks of motifs as a kind of "collective behavior." In this respect, the network motifs can be considered as "building blocks" of a network. So far, a few reports have appeared on how network motifs combine to form a larger structure (Dobrin et al. 2004; Kashtan et al. 2004; Yeger-Lotem et al. 2004). Their attentions are mainly on the topological aspects of the network. But more attention should be paid on functional aspects that cross talks between motifs give rise to (Kashtan et al. 2004).
In this paper, we focused our attention to this most prominent network motif, the feed-forward loop. At first, the stimulus–response profile of single feed-forward loop is analyzed. Secondly, we show that cross talking of feed-forward loops can give rise to a new feature of response that a single network motif alone cannot produce. These cross talks between the feed-forward loops are responsible for generating temporally multiple rise-and-fall expression pattern and spatially multiple stripes. It is then shown that the cross talks are found in the regulatory networks of E. coli, Saccharomyces cerevisiae, Strongylocentrotus purpuratus, and Drosophila melanogaster.
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Results |
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We consider the stimulus–response profile of single feed-forward loop as the dependence of Z expression on the level of transcription factor X. The feed-forward loops are classified into two types. When both direct and indirect pathways from X to Z are either positive or negative (same effects or synergistic), it is called "coherent" feed-forward loop; when the effects of both pathways are both positive and negative (opposing effects or antagonistic), it is called "incoherent" feed-forward loop (Shen-Orr et al. 2002). In a coherent feed-forward loop, the level of Z monotonically increases or decreases as the level of X increases, whereas in an incoherent feed-forward loop, it shows nonmonotonic and bell-shaped (convex or concave) responses (Fig. 2).
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The expression profile of Z can depend on the cis-regulation function of Z. Let us consider a case when the level of Z is activated by either X or Y (Fig. 2D). If interaction 1 is negative and interactions 2 and 3 are positive (Fig. 2C), the net effect of indirect pathway is negative as displayed in the red curve, and the direct regulatory one in the blue curve. Because Y is negatively controlled by X, the level of Z changes along the red curve in Fig. 2D. Therefore, the level of Z again exhibits a bell-shaped (concave) profile (Fig. 2C). As shown in these examples, the bell-shaped (convex or concave) profile with a single extreme point is a general and robust characteristic of incoherent feed-forward loops. The condition to obtain a bell-shaped response is that an activation threshold should be smaller (larger) than a repression threshold if the level of Z is not activated (is activated) when X product is absent (see Appendix for mathematical condition to obtain bell-shaped responses).
Cross talk of feed-forward loops generates a response with multiple peaks
When two network motifs share the same gene product (node), the two motifs are considered to be interacting or cross talking. Between two feed-forward loops, 12 types of interactions are possible (not shown). Among them, we found that five networks (Fig. 3A,C,E) exhibit stimulus–response profiles with target gene expressions that are activated within two intervals of stimulus concentration, whereas the rest cannot show such a response.
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The second example shown in Fig. 3C is also a network made up of two feed-forward loops. In this case, two products, Y and Z1, and the pathway between them are shared by both feed-forward loops. This network also shows a response with multiple peaks. For the cis-regulatory input functions of the genes encoding Z1 and Z2, we adopt the same type of function as is shown in Fig. 2C. In this network, X modulates the Z2 expression along three pathways, the two positive pathways on Z2 (XYZ2, and XZ1Z2), and the other negative pathway (XYZ1Z2). Each pathway introduces a threshold of activation or repression of Z2 in the concentration of X. Here, the threshold is defined as a concentration of X at which the concentration of Z2 is half of an extreme value. These thresholds determine the two intervals where the Z2 expression is elevated (Fig. 3D).
The networks shown in Fig. 3E are the same type as the network shown in Fig. 3C, in which two products and one pathway are shared by both the feed-forward loops. In these networks, the Z expression can be elevated also within two intervals in the concentration of X. When a feed-forward loop is added to a network in parallel, a new threshold of activation or repression of Z is introduced in the concentration of X so that the response of Z to the level of X exhibits a profile that is more complex.
A stimulus–response profile with a single or with multiple peaks is possible, if two or more thresholds of activation and repression exist in the stimulus concentration. Each threshold corresponds to one direct or indirect pathway from a regulatory gene to a target gene. Here, for simplicity we consider the case when a single pathway determines a single threshold value (a more complicated case, such a pathway which introduces several thresholds, may be possible when a cis-regulatory region is made up of two or more modules). The numbers of direct and indirect pathways from the most upstream regulatory factor to the most downstream target gene in a single feed-forward loop (Fig. 2), the series type (Fig. 3A), and the parallel type networks (Fig. 3C,E) are two, four, and three, respectively. Thus, these networks have two, four, and three thresholds, respectively, which determine intervals where the level of target gene expression is elevated.
Note that the examples shown here are not special kinds of networks that exhibit responses with multiple peaks. The multiple-peaks response is a robust property with respect to parameter variations, assuming the thresholds of activation and repression are ordered appropriately. For instance, in the case of Fig. 3D, the repression threshold must be set between the two activation thresholds (see Appendix for mathematical condition to obtain responses with a single and multiple peaks bounded by sharp thresholds).
Inferring regulatory pathways from multiple-peak responses
Based on this correspondence between a pathway and a threshold, one can infer regulatory pathways from experimental data. Once a response with multiple-peak profile has been obtained, one can estimate the minimum number of pathways of activation and repression between the two components, which is the number of thresholds in the stimulus–response profile. For instance, if a database about interactions between genes and components is available, one may infer the pathways that are responsible for the responses, and may indicate that there should be unknown components if the number of known pathways is smaller than the expected value. It is also suggested that if a response with multiple peaks performs a function, it is expected that many feed-forward loops are involved in the network.
Feed-forward loops work as "concentration detectors"
In a network composed of one or more feed-forward loops, when a stimulus concentration is lying within an interval between thresholds of activation and repression, the target gene expression is activated; otherwise, the gene expression is repressed. Thus, the network can work as a kind of "concentration detector." Such a concentration detection mechanism can perform various kinds of temporal and spatial information processing functions.
One biological function is temporal pulse generation (Basu et al. 2004). When the signal to the network increases with time, the target gene expression is transiently elevated when the signal concentration is lying between the thresholds. As an example, we studied temporal response of a network that consists of three feed-forward loops connected in a series (Fig. 4A). Consider the case where the level of X is 0 at time t = 0, and then X increases exponentially with time until it reaches a saturating concentration X = X f. In Fig. 4B, the time course of the levels of X, Z1, Z2, and Z3 is plotted (see Methods for the equations describing this temporal evolution of expression levels). As the level of X increases with time, genes Z1, Z2, and Z3 exhibit one, two, and four temporal pulses, respectively.
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If a temporal rise-and-fall profile or spatial stripe pattern is obtained experimentally, a regulatory factor may exist in the upstream components, which shows temporally or spatially a monotonic increasing profile of the concentration. For instance, in Fig. 4B,C, the upstream gene X exhibits a monotonic increasing profile, whereas Z1, Z2, and Z3 exhibit temporal rise-and-fall profiles and spatial stripes patterns, respectively. From these temporal and spatial profiles, the stimulus–response profiles can be reconstructed by plotting the concentrations of Z1, Z2, and Z3 against the level of X for each time point, and each location, respectively. From this reconstruction of the stimulus–response profile, the network responsible for generating these patterns can be inferred, if it is not known completely.
Frequency of cross talks and possibility of multiple-peak responses in transcriptional regulatory networks
The question now arises whether feed-forward loops form cross talks and the series and parallel types of interactions actually exist in real transcriptional regulatory networks. To investigate this, we performed a database analysis of the regulatory networks in E. coli, S. cerevisiae, S.purpuratus and D. melanogaster.
In these four species, the number of feed-forward loops is much larger than the expected values of randomized networks (Table 1), indicating that the loop is indeed considered as a network motif. Here, for the calculation of the expected value, see Procedures. The feed-forward loops found in the data sets of E. coli and D. melanogaster, shown in Fig. 5, are interacting and cross talking (Dobrin et al. 2004, Kashtan et al. 2004). The number of feed-forward loops that are separated from the others is only one in E. coli, and none in S. cerevisiae, S. purpuratus, and D. melanogaster. We enumerated the frequency with which two loops share genes in the four species. The frequency is defined as the ratio between the number of pairs that share genes and the number of possible combination of any two loops. The frequency tends to increase with the genome size (Table 1).
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Discussion |
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Note that in this paper we have limited our targets within transcriptional regulations. However, the consideration can be extended into other regulations involved in translation, post-translation, signal transduction, and so on without much complication.
The difference in the structure of networks may be related to their roles
Both in E. coli and in D. melanogaster, most feed-forward loops share genes and regulations. The structures of networks are, however, different between these species, which could relate to the difference in the function of the networks. In E. coli, the feed-forward loops may play roles for quick and coordinated responses against environmental variations (Kalir & Alon 2004). In contrast, because there are many networks of the series type and the parallel type in the regulatory network of D. melanogaster (Fig. 6B), the responses with multiple peaks may play roles in the species. In the early developmental stage of D. melanogaster, the maternal factors have formed gradients in the embryo and some of the downstream genes are going to exhibit stripe expression patterns. Our results suggest that the feed-forward loops may contribute to the pattern formation during early development.
Conditions of a response with multiple peaks
It is not obvious whether multiple thresholds can be set in a stimulus concentration of a target gene and multiple peaks are generated in vivo temporally or spatially. From a theoretical point of view, the following conditions can be pointed out to have multiple peaks. First, the activation and repression thresholds, each of which is determined by a direct or indirect pathway, should appear appropriately on the concentration coordinate, i.e. as the stimulus concentration increases, for instance, in the profile shown in Fig. 3D, first activation threshold appears, then repression, and finally activation. Otherwise, two activation intervals are merged into a single interval. To obtain sufficiently sharp thresholds, successive activation and repression thresholds are sufficiently deviated and the Hill coefficients around the thresholds should be large. Finally, the stimulus concentration changes over these thresholds under physiological condition.
Implication of feed-forward loops in pattern formation
Several examples are now known, where gene expression depends on morphogen concentration, and in a morphogen gradient, activation of the gene is induced within a spatial interval that is set by response thresholds of morphogen concentration (for review, see Gurdon & Bourillot 2001). For instance, in Xenopus blastula cells, the gene, Xbrachyury (Xbra) responses to a morphogen activin in a concentration range bounded by thresholds (Green et al. 1992, 1994; Gurdon et al. 1994). It was also indicated that the gene, Xbra might respond to two different concentration ranges of activin (Green et al. 1992, 1994). In these examples, incoherent feed-forward loops and their interactions may play roles to generate these patterns.
In the case of D. melanogaster, it is not completely clear if multiple stripes of a gene product are determined by a multiple-thresholds mechanism that we presented in this paper. Here, we want to propose that a parallel-type interaction of two feed-forward loops may play a role in generating the two stripes of gene giant (gt) by interpreting the anterior to posterior gradient of Hunchback (Hb) protein formed in the early stage (Driever & Nüsslein-Volhard 1989). In the first feed-forward loop, Hb represses both knirps (kni) and Krüppel (Kr) directly, and the product of kni represses Kr (Jäckle et al. 1986; Hoch et al. 1992; Wu et al. 2001; Jaeger et al. 2004a,b). As a result, Kr forms a stripe, and the borders are determined by the two pathways from hb to Kr (hbKr, and hbkniKr). Because Kr represses gt, the two pathways (hbKrgt and hbkniKrgt) determine the two borders of gt domains; the posterior border of anterior domain and the anterior border of posterior domain (Eldon & Pirrotta 1991; Kraut & Levine 1991a). As far as we know, there is no report concerning the factor that determines the anterior border of anterior gt domain. One candidate is hb, because it actually represses gt in the posterior region (Eldon & Pirrotta 1991; Kraut & Levine 1991a). In this case, hb, kni, Kr, and gt construct one of the parallel type networks shown in Fig. 3E (but each direct pathway is negative), and the three direct and indirect pathways from hb to gt establish the three borders of gt domains. Another candidate is tailless (tll), which is activated by Bicoid (Bcd) in the anterior tip region (Pignoni et al. 1992; Liaw & Lengyel 1993), and known as a repressor of gt in the posterior region of the embryo (Eldon & Pirrotta 1991; Kraut & Levine 1991a). In this case, the network that determines the borders of gt domains may contain other genes and pathways including the activation of hb by Bcd.
Most recently, in order to test the idea that a network can generate a stripe pattern by applying a concentration gradient, a synthetic experiment has been performed in vitro by engineering a gene circuit. Isalan et al. (2005) designed a network that consists of three gene products, A, B, and C, whose expression is controlled by transcription with T7 and SP6 RNA polymerases. From our viewpoint, one of the networks they made is essentially the same as shown in Fig. 3C. Among these five components, four, A, B, and C and T7 polymerase, form two feed-forward loops connected in parallel that are responsible for the stripe-pattern formation. They expected that the concentration of T7 polymerase forms a spatial gradient. The network interprets the gradient, forming the stripe pattern. They also introduced cross repressions between genes. Although the increase of repression interaction lowered the overall protein production, they reported that the thresholds became sharper.
Cross repressions between genes are actually observed in the gene regulatory network of D. melanogaster. For instance, the factor gt, which is inhibited by Kr (see previous discussion), also represses Kr (Kraut & Levine 1991b; Jaeger et al. 2004a,b). Such cross repression is contained in two types of network motifs made up of three genes, X, Y, and Z. In addition to the three interactions contained in a feed-forward loop, one contains an interaction from Y to X, and the other one contains an interaction from Z to Y, i.e. a feed-forward loop with cross-repression between Y and X, or Z to Y (Milo et al. 2004). A cross-repression between two genes sometimes gives rise to bistable genetic states as exemplified by genetic toggle switch (Gardner et al. 2000). In the case of the two motifs, however, as the level of X increases, bistable states are found only in a narrow concentration range, and the level of Z exhibits a bell-shaped profile with sharp thresholds against X concentration. Thus, we consider that a cross-repression introduced in a feed-forward loop is one of the ways to increase the Hill coefficient around a threshold and the threshold mechanism to generate a stripe pattern also works in these motifs (S.I. et al., manuscript in preparation). In the case of the stripe formation of gt, as Kr domain forms earlier than gt domain (Wu et al. 2001), Kr determines the domain of gt at first and then cross repression makes the boundary sharp.
Theoretical studies of morphogenesis
Wolpert (1969) proposed the model, in which the gradient of a morphogen conveys positional information and the cells interpret the concentration of morphogen to activate specific developmental programs. The subsequent question is how discrete thresholds are set within target cells in response to the gradient of morphogen. One possibility is to employ the concentration detection mechanism of feed-forward loops.
The theoretical study of morphogenesis was started by the pioneering work of Turing (1953). He proposed a reaction-diffusion mechanism that consists of two components, activator and inhibitor. These components form stripes and spot patterns spontaneously without any preceding gradient (Kondo & Asai 1995; Meinhardt & Gierer 2000). A uniform distribution at an initial stage is spontaneously destabilized and then a non-uniform distribution is evolved. One characteristic of this mechanism is that stripes have the intrinsic width. Therefore, the number of stripes is linearly proportional to the length of the system with stripes. In contrast, according to the concentration detection mechanism of feed-forward loops, a gradient of morphogen is indispensable. The formation of a pattern is a consequence of a response to a morphogen concentration at every location. Therefore, the number of stripes is maintained against the variations in the length of the system.
Outlook
There are several kinds of network motifs such as motifs with cross repression as well as feed-forward loop (Lee et al. 2002). Why do the regulatory networks contain particular structural elements so frequently? The reason may be related to their biological roles. In addition to the functions of individual network motifs, the interaction between them could be important for the motifs to play the roles. It would be interesting to explore what kinds of temporal and spatial signal processing abilities are possible by networks composed of several types of motifs with their interactions.
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Procedures |
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When gene Y is regulated by transcription factor X, the steady state level of Y expression is described by a function of the concentration of X. Let X and Y be the respective levels of X and Y expression, the level of Y expression is written by Y = f (X), where f(X) is a regulatory function (McAdams & Arkin 1998; Smolen et al. 2000; Mangan & Alon 2003). The time evolution of the level of Y is written by an ordinary differential equation:
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where 
is the depletion or dilution rate of the product Y.
The essential characteristic of the regulatory function f is described by the Hill equation: for positive control (activation):
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and for negative control (repression):
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where K is the Michaelis constant, H is the Hill coefficient, and V is the maximum expression level.
Cis-regulatory input functions
The level of Z is determined by both X and Y, Z = G(X,Y), where G(X,Y) is a cis-regulatory input function. Then, the time evolution of the level of Z is written by an ordinary differential equation:
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where is the depletion or dilution rate of the product Y.
The function, G can show complex dependence on the concentration X and Y (Buchler et al. 2003; Setty et al. 2003). Here, we do not aim to present a model that accounts for the detailed biochemistry of particular cis-regulations, but a toy model that captures the essential characteristics. In most cases, the function is essentially described by a combination of the regulatory functions, A(X) and R(X). In this paper, we adopted two types of cis-regulatory input function as examples.
For the first case, Y activates the Z expression, and the activation by Y is inhibited by X. The respective cis-regulatory function is written as the product of activation and repression regulatory functions:
G(X,Y) = R(X) x A(Y)
where R(X) is the repression regulation by X, and A(Y) is the activation regulation by Y. The parameters K, H, and V are suitably chosen for each binding of X and Y. The Michaelis constant K for the repression by X should be larger than K for the activation by Y; otherwise, the Z expression cannot be activated by Y. As is displayed in Fig. 2B, the Z expression is activated when the concentrations of Y are high and that of X are low enough. Such a cis-regulatory input function is found in one of the best characterized regulatory region lac operon of E. coli, which is regulated by the activator CAP, and the repressor LacI. The lac operon is activated only when LacI repressor is dissociated and activator CAP binds to the activator site.
For the second case, either X and Y can activate Z expression. The cis-regulatory input function G(X,Y) can be written as
G(X,Y) = A(X + Y)
which corresponds to the case that X and Y exclusively (competitively) bind to the same activation binding site. As is displayed in Fig. 2D, the Z expression is activated when either of the concentrations of X and Y are high. Such a cis-regulatory input function is found in the fliL promoter of E. coli, which is regulated by FliA and FlhDC (Kalir & Alon 2004).
Equations of feed-forward loops and parameters
Using the previously mentioned regulatory function A(X) and R(X) and cis-regulatory functions G(X,Y), the expression level of Z of a feed-forward loop can be written as a function of the X concentration X
Z = G(X, f (X))
where the function of f (X) is either A(X) or R(X). For the incoherent feed-forward loop shown in Fig. 2A, the level of Z expression is written by
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where A1, R2, and A3 are the regulation functions for interactions 1, 2, and 3 respectively. The parameters we used in Fig. 2A are given as V1 = 10.0, K1 = 10.0, H1 = 1.0, V2 = 1.0, K2 = 2.0, H2 = 2.0, V3 = 1.0, K3 = 0.1, and H3 = 2.0. Here, Vi, Ki , and Hi are the parameters of the function, Ai(X) or Ri(X). For the feed-forward loop shown in Fig. 2C, the level of Z expression is written by
Z = AZ(X + R1(X)).
The parameters used in Fig. 2C are given as V1 = 1.0, K1 = 0.002, and H1 = 1.0 for R1(X), and VZ = 1.0, KZ = 0.2, and HZ = 2.0 for AZ(X).
The extension of this expression to networks of several loops can be carried out by using the regulatory functions, A(X) and R(X), and pertinent cis-regulatory functions G(X,Y). For instance, the network of two loops shown in Fig. 3A, the level of Z1 and Z2 is given by
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where AYX and RYX are the regulatory function for the regulation on gene Y by X. The parameters used in Fig. 3A are given as 
= 5.0, 
= 1.0, 
= 2.0, 
= 1.0, 
= 2.0, 
= 3.0, 
= 30.0, 
= 2.0, 
= 2.0, 
= 4.0, 
= 0.5, 
= 2.0, 
= 1.0, 
= 5.0, 
= 3.0, 
= 1.0, 
= 2.0, and 
= 2.0. Here each parameter corresponds to the function with the same subscript. For the network of two loops shown in Fig. 3C, the level of Z2 expression is given by
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where A YX and RYX are the regulatory function for the regulation on gene Y by X. In Fig. 3C, the parameters for the functions are given as VYX = 50.0, KYX = 5.0, HYX = 2.0, = 1.0, = 10.0, = 4.0, 
= 1.0, 
= 5.0, 
= 2.0, = 1.0, 
= 0.2, 
= 2.0, 
= 1.0, 
= 0.1, and 
= 4.0, where each parameter corresponds to the function with the same subscript.
Temporal and spatial responses of feed-forward loops
In order to study temporal and spatial responses of the network shown in Fig. 4A, we use the following equations and parameters values.
The temporal response of the network is described by the following ordinary differential equations. For transcription factor X, the concentration of X at time t = 0, X(t) = 0, and the temporal evolution is described by
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where X is the depletion rate and X f is the saturating concentration. The feed-forward loops are described by a set of equations written by
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where 
and 
are the depletion rate of Yi and Zi, respectively, Ai (Zi –1) is the activation regulatory function for gene Yi, and Gi (Zi –1,Yi ) is the cis-regulatory function of gene Zi. For the cis-regulatory function, we adopt the function shown in Fig. 2B, i.e. Gi (Zi –1,Yi ) = Ri (Zi –1)Ai(Yi ). For Fig. 4B, the expression level of X develops with the parameters, X = 1.0 x 10–3, Xf = 0.5. The parameters for all Yi(i = 1, 2, 3) are given as 
= 1.0 and 
= 0.08 in the function Ai (Zi–1) and 
for the depletion rate. The parameters for all Zi(i = 1, 2, 3) are given as 
= 2.0, 
= 0.1 and 
= 0.1 in the function Gi(Zi–1,Yi), and 
= 1.0 for the depletion rate.
The spatial expression profile shown in Fig. 4C is the solution of the following partial differential equations. For X, the translation of X takes place only at the localized area in the anterior pole (0 < l < l 0) at the rate of SX. Then the spatiotemporal expression profile follows the equation given by
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where the first term describes the depletion process, and the second term is the diffusion process. Furthermore, the term SX is added between the range 0 < l < l 0, representing the translation of X. The steady state solution of this equation exhibits the exponential profile shown in Fig. 4C. The expression profile of the feed-forward loops is described by a set of equations given by
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where the first terms on the right hand sides are the same as the case of temporal response, and the second term describes the diffusion process. For Fig. 4C, we used the same parameters as in Fig. 4B for X (= 
), , , , , 
, , and (i = 1, 2, 3). All the diffusion coefficients are set as DX = 
= 
= 10.0 (i = 1, 2, 3). The system size is set as L = 100.0, and the translation of X takes place in the region 0 < l < 1.0 with rate SX = 1.0.
Transcription network database
For the bacterium, E. coli, we used the database released by and collaborators (2002) (http://www.weizmann.ac.il/mcb/UriAlon), consisting of operons regulated by transcription factors. The data set contains 423 genes and 578 regulations. For the yeast, S. cerevisiae, we used the literature-based database YPD (yeast protein database) containing directed positive or negative interactions (Costanzo et al. 2001) (http://www.proteome.com/YPDhome.html). The data set contains 807 genes and 862 regulations. For the fruit fly, D. melanogaster, we used the literature-based database GeNet (Selov et al. 1998) (http://www.csa.ru/Institute/gorb_Department/inbios/genet/genet.htm), containing the transcription interactions involved in early development. Because the original data contains many ambiguous interactions, we chose regulatory paths indicated as "direct" or "possibly direct" in the original data. The data set contains 122 genes and 309 regulations. We also analyzed the transcription interactions of the sea urchin S. purpuratus, which govern the endomesoderm specification (Oliveri & Davidson 2004) (http://www.its.caltech.edu/mirsky/), although the size of the data set may still not be large enough to determine the large-scale distribution of network motifs. The data set contains 43 genes and 87 regulations.
In order to study the abundance of feed-forward loops, we compared the number of feed-forward loops that appeared and the expected number of the loops in a set of randomized network. The set of networks is generated by randomizing the connections in a real network without changing the numbers of incoming and outgoing arrows of each node as well as the numbers of nodes and arrows. Then, the expected value is obtained by averaging the number of the feed-forward loops that appeared in each network over the set. If the number of the loops is much larger than the expected number, we consider it as a network motif (Milo et al. 2002; Shen-Orr et al. 2002).
To determine the number of incoherent feed-forward loops, the regulations that are not specified as activation or repression were excluded. Then, the numbers of feed-forward loops and incoherent feed-forward loops were counted again. The frequency of feed-forward loops that share genes and interactions with other feed-forward loops was calculated as the ratio between the number of such feed-forward loops and the total number of combinations between two loops.
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Appendix |
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The activation threshold is determined by the overall effect of indirect pathway mediated by Y. When the concentration of Y reaches the Michaelis constant KYZ of interaction 3, the strength of activation of Z by Y reaches half of the maximum value. The activation threshold is given by the concentration of X at which the concentration of Y reaches KYZ. As shown in Equation 1 in the Procedures section, the expression level of Y is given by
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where VY is the maximum expression level of Y, and KXY is the Michaelis constant, and HXY is the Hill coefficient of interaction 1. When the concentration of Y reaches KYZ, the concentration X is given by KXYZ:
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This expression gives the activation threshold of Z expression. In order for Z to be activated, KYZ must be smaller than VY; otherwise, the activation regulation by X does not function. The indirect negative control of Z by X mediated by Y is effectively described by the activation regulatory function A(X), shown in Equation 1, in which KXYZ gives the Michaelis constant and HXYZ = HXYHYZ is the Hill coefficient in the function.
The repression threshold is determined by the Michaelis constant KXZ of the direct interaction 2. When the concentration of X is KXZ, the strength of repression by X reaches half of the maximum value.
In the feed-forward loops, as the level of X increases, Z is activated approximately at X = KXYZ and repressed at X = KXZ, and the activation interval of Z is described by these two concentrations. Therefore, the condition KXYZ< KXZ must be satisfied for Z to be activated and thus to exhibit a bell-shaped expression profile.
The sensitivity of Z to the change in the concentration of X is characterized by a Hill coefficient. For a bell-shaped profile, two Hill coefficients are defined: one is for the intervals indicating increasing profile, and the other one is for the intervals showing decreasing profile. These coefficients are approximately given by HXYZ for increasing profile (activation) and HXZ for decreasing profile (repression).
The maximum expression level of Z should be sufficiently strong; otherwise, the bell-shaped profile does not function. Suppose that the indirect pathway is approximately and effectively characterized by a Hill equation with Michaelis constant KXYZ and Hill coefficient HXYZ. Then, the expression level of Z, given by Equation 3, is approximately written by
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We further suppose that the Hill coefficients, HXYZ and HXZ are the same value, H. Then, the maximum value of Z expression is given by
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Thus, under the condition KXYZ< KXZ, the maximum value Zmax increases with H and KXZ/KXYZ. Even when HXYZ and HXZ take different values, the dependence of Zmax on the parameters is not qualitatively different.
This consideration is extended to the profile with multiple peaks. As an example, we study the network shown in Fig. 3D. There are three thresholds, K1, K2, and K3, each of which corresponds to a particular pathway from X to Z2 in Fig. 3C. Suppose that K1 and K3 are activation thresholds (K1 < K3), and K2 is the repression threshold. To have the two distinct intervals where the level of Z is elevated as shown in Fig. 3D, the condition K1 < K2 < K3 should be satisfied. Around the concentration of each threshold, a Hill coefficient can be defined. A profile with sharp peaks is obtained when the Hill coefficients are sufficiently large.
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Acknowledgements |
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Footnotes |
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* Correspondence: E-mail: shibata{at}hiroshima-u.ac.jp
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References |
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